Question

\(lim\left(x->2\right)=\dfrac{\sqrt{x+2}-x}{\sqrt{4x+1}-3}\)

Answer 1

\(=\lim\limits_{x\rightarrow2}\dfrac{\sqrt{x+2}-2+2-x}{\dfrac{\left(\sqrt{4x+1}-3\right)\left(\sqrt{4x+1}+3\right)}{\sqrt{4x+1}+3}}=\lim\limits_{x\rightarrow2}\dfrac{\left(\sqrt{4x+1}+3\right)\left(\dfrac{x-2}{\sqrt{x+2}+2}-\left(x-2\right)\right)}{2\left(x-2\right)}\)

\(=\lim\limits_{x\rightarrow2}\dfrac{\left(\sqrt{4x+1}+3\right)\left(\dfrac{1}{\sqrt{x+2}+2}-1\right)}{2}=-\dfrac{9}{4}\)